Recently, a horizon-independent pricing approach [1]} is proposed for the ad auctions in which the seller does not know in advance the number of played rounds, and Mirrokni et al. [2]} designed the new contract for advertisers with advanced targeting capabilities in order to maximize the revenue. Budget management is considered in ad auctions by combining the seller's profit and buyers' utility [3]}. Many works discuss price design, including reserve prices design in single item auctions [4]}, mixture bidder of GSP and Vickrey-Clarke-Groves (VCG) [5]}, algorithmic/dynamic pricing [6]},
joint optimization of bid and budget allocation in sponsored search [7]}, optimal real-time bidding for display advertising [8]}, budget constrained bidding in keyword auctions and online knapsack problems [9]}, etc. The exploration-exploitation strategy has been used in contextual multi-armed bandit settings for content recommendation [10]}, with further extension to cluster-of-bandit algorithm [11]} for collaborative recommendation tasks, and linear bandit algorithm [12]}. Orthogonal to the above works [13]}, [14]},  [15]}, this paper presents the constraint bid optimization
for the best conversion performance while achieving the CPA goals. The short version of this work appeared in  [16]}.
